THEORY OF THE POTENTIOMETRIC CURVE OF AN AMALGAM OF TWO METALS FORMING A SOLUBLE COMPOUND WITH EACH OTHER

A. G. STROMBERG, Yu. P. BELOUSOV

In paper [1] a derivation was given of mathematical relations for the potentiometric curve of an amalgam of two metals forming with each other a sparingly soluble intermetallic compound in mercury.

The aim of the present work is to derive a theoretical equation for the potentiometric curve of an amalgam of two metals forming an intermetallic compound soluble in mercury, to investigate this equation, and to develop simple and reliable graphical methods for calculating the thermodynamic parameters of intermetallic compounds of any composition soluble in mercury.

Let us suppose that the metals \(M\) and \(N\) (metal \(M\) is the more electronegative in the system) form with each other in the amalgam a soluble intermetallic compound of composition \(MN_m\), which dissociates according to the equation\(^1\)

\[ MN_m \rightleftarrows M + mN . \tag{1} \]

The expression for the instability constant of the intermetallic compound \(K_{\mathrm{н}}\) has the form

\[ K_{\mathrm{н}} = C'_1 \cdot (C'_2)^m / C'_3 \quad \text{or} \quad K_{\mathrm{н}}(C_1 - C'_1)/C'_1 = [C_2 - m(C_1 - C'_1)]^m, \tag{2} \]

where \(C_1\) and \(C_2\) are the analytical concentrations of metals \(M\) and \(N\) in the amalgam; \(C'_2\), \(C'_1\), \(C'_3\) are the equilibrium concentrations in the amalgam, respectively, of \(N\), \(M\), and the intermetallic compound \(MN_m\). In doing so we assume that the activity coefficients of the metals in the investigated concentration interval of metal \(M\) are approximately constant and include them in the numerical value of the instability constant \(K_{\mathrm{н}}\).

We introduce the dimensionless variables \(\alpha\) and \(\beta\), which are related to the experimentally determined quantities \(E\), \(C_1\), and \(C_2\) by the relations:

\[ \alpha = C'_1/C_1,\quad -\lg \alpha = \varepsilon = (E - E_0)zF/2{,}3RT,\quad \beta = C_2/C_1,\quad \lg \beta = \delta, \tag{3} \]

where \(E_0\) and \(E\) are the emf of the concentration amalgam cell, respectively in the absence and in the presence of metal \(N\) in the amalgam; \(z\) is the number of electrons participating in the potential-determining electrode reaction.

Substituting the values of \(\alpha\) and \(\beta\) from relations (3) into expression (2), we obtain the equation of the potentiometric curve of an amalgam of two metals forming with each other an intermetallic compound soluble in mercury,

\[ B(1-\alpha)/\alpha = [1 - m(1-\alpha)/\beta]^m, \tag{4} \]

\(^1\) If the intermetallic compound has the composition \(M_aN_b\), then it must be represented in the form \(MN_m\), where \(m = b/a\).

where

\[ B = K_{\mathrm{н}}/C_2^m. \tag{5} \]

The advantage of this equation (4) is that it does not depend on the charge of the ion \(M^{z+}\), on the absolute values of the concentrations \(C_1\), \(C_2\), and \(\bar C_1\) (\(\bar C_1\) is the concentration of metal \(M\) in the auxiliary electrode). Knowing the concentration \(C_2\) and the parameters \(m\) and \(B\), formula (5) can be used to calculate easily the instability constant of the intermetallic compound, \(K_{\mathrm{н}}\).

We propose to carry out the mathematical investigation of equation (4) in the dimensionless variables \(\alpha\) and \(\beta\), and to make the graphical interpretation on a plot in the dimensionless coordinates \(\varepsilon,\delta\). Such a double system of notation, despite its apparent complexity at first glance, is justified. Substitution of the variables \(\varepsilon\) and \(\delta\) directly into equation (4) leads to its unnecessary complication, while representing the plot directly in the coordinates \(\alpha\) and \(\beta\) is inconvenient because these variables change by several orders of magnitude along the potentiometric curve.

Let us proceed to the investigation of equation (4) of the potentiometric curve. For \(\beta \to \infty\) (\(\delta \to \infty\)) we obtain

\[ B(1-\alpha_\infty)/\alpha_\infty = 1,\qquad \alpha_\infty = \frac{B}{B+1}, \tag{6} \]

\[ \varepsilon_\infty = \lg (B+1)/B. \]

Fig. 1. Theoretical potentiometric curve (curve 1) for a soluble intermetallic compound of composition \(MNm\) (\(m=1\)), calculated by formula (5) for the numerical value \(B=0.1\); curves 2 and 3 are the upper and left limiting branches of the curve, calculated by formulas (6) and (7); 4 is the asymptote of the left limiting branch

Fig. 2. Theoretical potentiometric curves (1–9) for nine soluble intermetallic compounds at three values of the parameter \(m\) (\(1/3\); 1; and 3) and at three values of the parameter \(B\) (0.1; 0.01; and 0.001), calculated by formula (4). Curves 10–12 are the three left-hand, and straight lines 13–15 the three upper, limiting branches of the curves. Straight lines 16–18 are asymptotes of the left limiting branches. The dotted lines indicate distances from the point on the curve to the left and upper limiting branches

Thus, as \(\delta \to \infty\), the value of \(\varepsilon\) tends to a certain limiting value \(\varepsilon_\infty\). On the plot in the coordinates \(\varepsilon,\delta\) (Fig. 1), this limiting value \(\varepsilon_\infty\) is represented by the horizontal straight line 2 (the upper limiting branch). From expression (6) it is evident that \(\varepsilon_\infty\) depends only on the parameter \(B\) and does not depend on the parameter \(m\). Therefore, a graphical estimate of the limiting value \(\varepsilon_\infty\) can serve for determination of the parameter \(B\).

For \(\alpha \to 1\) (\(\varepsilon \to 0\)), the quantity \((1-\alpha)\to 0\), and, consequently, from formula (4) we obtain the equation of the left limiting branch (curve 3 on

Fig. 1), which the potentiometric curve in the \(\varepsilon, \delta\) plot approaches at small \(\varepsilon\).

\[ \beta' = m(1-\alpha'); \quad \delta' = \lg(1-\alpha')+\lg m. \tag{7} \]

As \(\alpha \to 0\) (\(\varepsilon \to \infty\)), the left limiting branch asymptotically approaches the value \(\lim_{\varepsilon \to \infty}(\delta')=\lg m\). Thus, the position of asymptote 4 in the plot makes it possible to determine the parameter \(m\). In Fig. 2, in the coordinates \(\varepsilon, \delta\), nine theoretical curves are presented, calculated from equation (5) for three values of the parameter \(B\) and three values of \(m\).

Comparison with the preceding paper [1] shows that the \(\varepsilon, \delta\) plots for an intermetallic compound soluble and slightly soluble in mercury differ substantially from one another and may thus serve as a criterion for elucidating the phase nature of an intermetallic compound. If the curve in the \(\varepsilon, \delta\) plot, with increasing \(\delta\), tends toward a certain limiting value, this is an indication of an intermetallic compound soluble in mercury. If the curve, with increasing \(\delta\), passes through a maximum and again approaches the abscissa axis, this is an indication of an intermetallic compound slightly soluble in mercury.\(^2\) Let us note the consequences of equation (5) for the potentiometric curve and the plot in the coordinates \(\varepsilon, \delta\). For the equivalence point we have \(C_2=mC_1\) and \(\beta=m\) (\(\delta=\lg m\)); then from equation (5) we obtain the expression

\[ \alpha_e^{\,m+1}=B(1-\alpha_e) \quad \text{or} \quad (m+1)\varepsilon_e=-\lg B-\lg(1-\alpha_e). \tag{8} \]

Fig. 3. Theoretical plot in the coordinates \(\varepsilon_e, -\lg B\) at different values of the parameter \(m\), calculated from formula (8). The plot also shows the dependence of \(\varepsilon_\infty\) on \(-\lg B\) in accordance with formula (6).

Fig. 4. Theoretical plot in the coordinates \(g, -\lg B\) at different values of the parameter \(m\), calculated from formula (9). The dashed horizontal lines on the plot indicate the limiting values of \(g\) at small values of the parameter \(B\). Dashed straight line 5 indicates the limiting value of \(-\lg B\), at which \(g\) becomes practically a constant quantity.

From the graph in Fig. 1 it is seen that the equivalence point \(E\) lies at the intersection of curve 1 with asymptote 4 of limiting branch 3. The plot in the coordinates \(\varepsilon_e-\lg B\) at different values of the parameter \(m\), calculated according to the formula—

\(^2\) The difference in the forms of the potentiometric curve (in the plot in the coordinates \(E, \lg C_1/\overline{C}_1\)) for a soluble and slightly soluble intermetallic compound, for the particular case \(m=1\), was first indicated in [2].

where (8) is presented in Fig. 3. It is seen from the graph that, from the magnitude of the ordinate \(\varepsilon_e\) at the equivalence point, knowing the parameter \(m\), one can determine the parameter \(B\).

On curve 1 (Fig. 1) there is a point \(D\) at which the deviations from both limiting branches are identical and equal to \(g\). This deviation, in accordance with equation (4), is related to the parameters \(B\) and \(m\) by the relation

\[ B=\frac{1/a-[(a-1)/a]^m}{[(a-1)/a]}, \tag{9} \]

where

\[ g=\lg a. \]

Figure 4 gives a plot in the coordinates \(g, -\lg B\) for different values of the parameter \(m\), calculated from equation (9). For sufficiently small values of the parameter \(B\) (for large values of \(-\lg B\)), the equal deviation \(g\) becomes constant for each \(m\). Thus, by determining the value of \(g\) on an experimental graph in the coordinates \(\varepsilon, \delta\), one can find approximate values of \(m\) and \(B\), or \(K_{\mathrm{n}}\).

LITERATURE

1. A. G. Stromberg, B. P. Belousov. Elektrokhimiya, 1973, No. 3.
2. A. I. Zebreva. Proceedings of the Institute of Chemical Sciences, Academy of Sciences of the Kazakh SSR, 15, (54, 1967).